Four 4s Puzzle

Using exactly four of the digit "4" and any mathematical symbols you choose, for which natural numbers can you create a mathematical expression equal to that number? For example, 1 = 4 ÷ 4 + 4 - 4 . I would suggest trying it for the numbers 2 through 20 before reading on. If you're really adventuresome, try it for all of the numbers between 2 and 100 (and don't worry if you can't figure out some numbers, this will be discussed later).

Now that you've had a chance to try this out, let's look at the results.

What mathematical symbols did you use? Most likely you started with the basic arithmetical signs, but you may have used some other symbols as well. You may have found the following to be useful:

• +, the plus sign
• −, the minus sign
• ×, the multiplication symbol
• "÷" or "/", the division symbol
• "(" and ")", parentheses
• Abuttal. This isn't strictly a mathematical operator, but could be used according to the puzzle description. For example, 44 + 44 = 88.
• . or the decimal point. Similar to "abuttal" immediately above. For example, 4 ÷ .4 + 4 + 4 = 18.
• !, or the factorial operator.
• %, or the percent sign. Don't forget that percent simply means "divided by (per) one hundred (cent)."
• √, or the square root operator.
• Exponentiation. For example, 4^4−4÷4 = 64.
• What other symbols did you find useful? Feel free to contact me (see contact information at bottom of the page) if you found any others useful.

Next question: What numbers did you find the hardest to find expressions for? Most likely, you found odd numbers much harder to create than even numbers. It's not too hard to see why, looking at the properties of integers. With this puzzle, you start with even integers (specifically, "4"). Whenever you add two even integers, you get another even integer. If you subtract two even integers, you get another even integer. If you multiply an even integer by any integer, you get an even integer. If you take the square root of an even integer whose square root is not irrational, you get another even integer. If you divide two even integers, you may get an even integer, an odd integer, or a decimal (depending on each of the numbers' prime factorizations), but in order to get larger odd numbers, the dividend needs to be large to begin with.